Tree Structures

Abstract

The next step in preparing the completeness proof of the Q calculi is to show that any μ-hypercomplete set α is embeddable into a structure 〈W, R, Φ〉 satisfying certain conditions outlined in 10.0 already. As one would guess, we can assume here that 〈W, R〉 is a tree with the top w ₀∈ W, where Φ(w ₀)= α, our starting set. We shall compress the pair 〈W, R〉 into a partially ordered set Σ⊂ω, where Σ will be called an index tree (Section 13.1). Then we introduce the notion of μtree structures 〈Σ, Φ〉 where Φ is a function defined on Σ, and the values of Φ are μ-hypercomplete sets (Section 13.2). Finally, we show (in Section 13.3) the embeddability of α into a so-called complete μ-tree structure. (The omitted relation R can be defined by means of the partial ordering on Σ; but we shall need R only in the next §.)